Exponential-supermartingale escape bound

If $e^{-\theta N_{t\wedge {\sigma }_K}}$ is a nonnegative supermartingale, with ${\sigma }_K$ the first time $N_t\le K$, the optional stopping theorem yields ${\mathbb{P}}_N({\sigma }_K<\infty )\le e^{-\theta (N-K)}$. Such a bound can prove escape of a backlog independently of heuristic fluid models.